1 9 The Matrix of a Linear Transformation

1. 9 The Matrix of a Linear Transformation

Example: Suppose T is a linear transformation from such that and to . With no additional information, find a formula for the image of an arbitrary x in .

Theorem 10. Let T: a linear transformation. Then there exists a unique matrix A such that for a x in. In fact, A is the matrix whose jth column is the vector where is the jth column of the identity matrix in. Note: A is called the standard matrix for the linear transformation T.

Example: Find the standard matrix A for the dilation transformation T(x)=4 x, where

Find the standard matrix of each of the following transformations. Reflection through the x-axis Reflection through the y=x Reflection through the y=-x Reflection through the origin

Find the standard matrix of each of the following transformations. Horizontal Contraction & Expansion Vertical Contraction & Expansion Projection onto the x-axis Projection onto the y-axis

Definition A mapping if each b in is said to be onto is the image of at least one x in . Definition A mapping if each b in is said to be one-to-one is the image of at most one x in .

Theorem Let be a linear transformation. Then, T is one-to-one if and only if has only the trivial solution. Theorem Let be a linear transformation with the standard matrix A. 1. T is onto iff the columns of A span. 2. T is one-to-one iff the columns of A are linearly independent